The electric and magnetic fields along with the electric and
magnetic vector potentials are expanded in the
\emph{normal} and \emph{transverse} coordinate system as shown below.
\begin{align}
\vec{H}&=\vec{H}_\bot+H_n\hat{n}\label{eqn:Hexpnt}\\
\vec{E}&=\vec{E}_\bot+E_n\hat{n}\label{eqn:Eexpnt}\\
\vec{A}&=\vec{A}_\bot+A_n\hat{n}\label{eqn:Aexpnt}\\
\vec{F}&=\vec{F}_\bot+F_n\hat{n}\label{eqn:Fexpnt}
\end{align}
The operator $\vec{\nabla}$ is also separated into transverse
$(\bot)$ and normal $(n)$ components.
\begin{equation*}
    \vec{\nabla}=\vec{\nabla}_\bot+\frac{\partial}{\partial n}\hat{n}\eqno{(\ref{eqn:sepnabla})}
\end{equation*}
Expanding the Laplacian operator yields,
\begin{equation*}
\nabla^2=\vec{\nabla}\cdot\vec{\nabla}=\nabla_\bot^2+\frac{\partial^2}{{\partial
    n}^2}\eqno{(\ref{eqn:sepnablasqr})}
\end{equation*}


\subsection{Normal and Transverse Fields from Magnetic Vector Potential}
\begin{align}
\vec{H}^A_\bot&=\vec{\nabla}_\bot{A_n}\times\hat{n}+\hat{n}\times\frac{\partial\vec{A}_\bot}{\partial n}\label{eqn:HperpA0}\\
H^A_n&=\vec{\nabla}_\bot\times\vec{A}_\bot\cdot\hat{n}\label{eqn:HnormA0}
\end{align}
\begin{align}
\vec{E}^A_\bot&=\frac{1}{j\omega\varepsilon'}\biggl[\vec{\nabla}_\bot\bigl(\vec{\nabla}_\bot\cdot\vec{A}_\bot\bigr)+\vec{\nabla}_\bot\frac{\partial A_n}{\partial n}-\gamma^2 \vec{A}_\bot\biggr]\label{eqn:EperpA0}
\end{align}
\begin{align}
\vec{E}_n^A&=\frac{1}{j\omega\varepsilon'}\biggl[\frac{\partial}{\partial{n}}\nabla_\bot\cdot\vec{A}_\bot+\frac{\partial^2{A}_n}{\partial{n}^2}-\gamma^2A_n\biggr]\label{eqn:EnormA0a}\\
&=\frac{1}{j\omega\varepsilon'}\biggl[\frac{\partial}{\partial{n}}\nabla_\bot\cdot\vec{A}_\bot-\gamma_\bot^2A_n\biggr]\label{eqn:EnormA0b}
\end{align}

\subsection{Normal and Transverse Fields from Electric Vector Potential}
\begin{align}
\vec{E}^F_\bot&=-\vec{\nabla}_\bot{F_n}\times\hat{n}-\hat{n}\times\frac{\partial\vec{F}_\bot}{\partial n}\label{eqn:HperpF0}\\
E^F_n&=-\vec{\nabla}_\bot\times\vec{F}_\bot\cdot\hat{n}\label{eqn:HnormF0}\\
\vec{H}^F_\bot&=\frac{1}{j\omega\mu'}\biggl[\vec{\nabla}_\bot\bigl(\vec{\nabla}_\bot\cdot\vec{F}_\bot\bigr)+\vec{\nabla}_\bot\frac{\partial F_n}{\partial n}-\gamma^2 \vec{F}_\bot\biggr]\label{eqn:EperpF0}\\
\vec{H}_n^F&=\frac{1}{j\omega\mu'}\biggl[\frac{\partial}{\partial{n}}\nabla_\bot\cdot\vec{F}_\bot+\frac{\partial^2{F}_n}{\partial{n}^2}-\gamma^2F_n\biggr]\label{eqn:EnormF0a}\\
&=\frac{1}{j\omega\mu'}\biggl[\frac{\partial}{\partial{n}}\nabla_\bot\cdot\vec{F}_\bot-\gamma_\bot^2F_n\biggr]\label{eqn:EnormF0b}
\end{align}

\subsection{Normal and Transverse Fields from Vector Potentials}
\begin{align}
E_n&=-\vec{\nabla}_\bot\times\vec{F}_\bot\cdot\hat{n}+\frac{1}{j\omega\varepsilon'}\biggl[\frac{\partial}{\partial{n}}\nabla_\bot\cdot\vec{A}_\bot-\gamma_\bot^2A_n\biggr]\label{eqn:Enorm0}\\
H_n&=\vec{\nabla}_\bot\times\vec{A}_\bot\cdot\hat{n}+\frac{1}{j\omega\mu'}\biggl[\frac{\partial}{\partial{n}}\nabla_\bot\cdot\vec{F}_\bot-\gamma_\bot^2F_n\biggr]\label{eqn:Hnorm0}\\
\vec{E}_\bot&=\frac{1}{j\omega\varepsilon'}\biggl[\vec{\nabla}_\bot\bigl(\vec{\nabla}_\bot\cdot\vec{A}_\bot\bigr)+\vec{\nabla}_\bot\frac{\partial A_n}{\partial n}-\gamma^2 \vec{A}_\bot\biggr]-\vec{\nabla}_\bot{F_n}\times\hat{n}-\hat{n}\times\frac{\partial\vec{F}_\bot}{\partial n}\label{eqn:Eperp0}\\
\vec{H}_\bot&=\frac{1}{j\omega\mu'}\biggl[\vec{\nabla}_\bot\bigl(\vec{\nabla}_\bot\cdot\vec{F}_\bot\bigr)+\vec{\nabla}_\bot\frac{\partial F_n}{\partial n}-\gamma^2 \vec{F}_\bot\biggr]+\vec{\nabla}_\bot{A_n}\times\hat{n}+\hat{n}\times\frac{\partial\vec{A}_\bot}{\partial n}\label{eqn:Hperp0}
\end{align}

\subsection{Hybrid Solutions $(E_n\ne0,H_n\ne0)$}
Let the transverse vector potentials be equal to zero,
\begin{align}
	\vec{F}_\bot = \vec{A}_\bot = 0
\end{align}
then (\ref{eqn:Enorm0})--(\ref{eqn:Hperp0}) reduce too,
\begin{align}
E_n&=\frac{-(\gamma_\bot^2=-k_\bot^2)}{j\omega\varepsilon'}A_n\label{eqn:Enorm1}\\
H_n&=\frac{-(\gamma_\bot^2=-k_\bot^2)}{j\omega\mu'}F_n\label{eqn:Hnorm1}\\
\vec{E}_\bot&=\frac{1}{j\omega\varepsilon'}\vec{\nabla}_\bot\frac{\partial A_n}{\partial n}-\vec{\nabla}_\bot{F_n}\times\hat{n}\label{eqn:Eperp1}\\
\vec{H}_\bot&=\frac{1}{j\omega\mu'}\vec{\nabla}_\bot\frac{\partial F_n}{\partial n}+\vec{\nabla}_\bot{A_n}\times\hat{n}\label{eqn:Hperp1}
\end{align}
Since $E_n\propto{A}_n$ and $H_n\propto{F}_n$ then by solving for $A_n$ and $F_n$ in (\ref{eqn:Enorm1}) and (\ref{eqn:Hnorm1}) and substituting into (\ref{eqn:Eperp1}) and (\ref{eqn:Hperp1}) gives the perpendicular electric and magnetic fields as a function of the normal electric and magnetic field components.  
\begin{align}
\vec{E}_\bot&=\frac{1}{k_\bot^2}\left[\vec{\nabla}_\bot\frac{\partial E_n}{\partial n}-j\omega\mu'\vec{\nabla}_\bot{H_n}\times\hat{n}\right]\label{eqn:Eperp2}\\
\vec{H}_\bot&=\frac{1}{k_\bot^2}\left[\vec{\nabla}_\bot\frac{\partial H_n}{\partial n}+j\omega\varepsilon'\vec{\nabla}_\bot{E_n}\times\hat{n}\right]\label{eqn:Hperp2}
\end{align}
These are seen to be identical to (\ref{eqn:TETMmodeSolntemp4}) and (\ref{eqn:TETMmodeSolntemp5}).

\subsection{$TEM_n$ Solutions $(E_n=0,H_n=0)$}
It is apparent by examining (\ref{eqn:Enorm0}) and (\ref{eqn:Hnorm0}) that in order to obtain $TEM_n$ solutions then the following conditions must be met.
\begin{align}
	\vec{F}_\bot = \vec{A}_\bot = \gamma_\bot^2 = 0
\end{align}
which makes $\gamma^2=\gamma_n^2$ from the constraint equation. The normal electric and magnetic fields are then found from (\ref{eqn:Enorm1}) amd (\ref{eqn:Hnorm1}) to be,
\begin{align}
E_n&=H_n=0\label{eqn:EHnormTEM}
\end{align}
After the above conditions are met any of the following cases can be used,
\begin{enumerate}
	\item $An\neq0$ and $F_n\neq0$ in which the transverse fields are found to be,
\begin{align}
\vec{E}_\bot&=\frac{1}{j\omega\varepsilon'}\vec{\nabla}_\bot\frac{\partial A_n}{\partial n}-\vec{\nabla}_\bot{F_n}\times\hat{n}\label{eqn:EperpTEM1}\\
\vec{H}_\bot&=\frac{1}{j\omega\mu'}\vec{\nabla}_\bot\frac{\partial F_n}{\partial n}+\vec{\nabla}_\bot{A_n}\times\hat{n}\label{eqn:HperpTEM1}
\end{align}
	\item $An=0$ and $F_n\neq0$ in which the transverse fields are found to be,
\begin{align}
\vec{E}_\bot&=-\vec{\nabla}_\bot{F_n}\times\hat{n}\label{eqn:EperpTEM2}\\
\vec{H}_\bot&=\frac{1}{j\omega\mu'}\vec{\nabla}_\bot\frac{\partial F_n}{\partial n}\label{eqn:HperpTEM2}
\end{align}
	\item $An\neq0$ and $F_n=0$ in which the transverse fields are found to be,
\begin{align}
\vec{E}_\bot&=\frac{1}{j\omega\varepsilon'}\vec{\nabla}_\bot\frac{\partial A_n}{\partial n}\label{eqn:EperpTEM3}\\
\vec{H}_\bot&=\vec{\nabla}_\bot{A_n}\times\hat{n}\label{eqn:HperpTEM3}
\end{align}
\end{enumerate}

 
\subsection{$TM_n$ Solutions $(H_n=0)$}
It is apparent by examining (\ref{eqn:Enorm0}) and (\ref{eqn:Hnorm0}) that in order to obtain $TM_n$ solutions then the following conditions must be met.
\begin{align}
	\vec{F} = 0\\
	\vec{A} = A_n\hat{n}
\end{align}
and then the fields are found form (\ref{eqn:Enorm1})--(\ref{eqn:Hperp1}) to be,
\begin{align}
E_n&=\frac{-(\gamma_\bot^2=-k_\bot^2)}{j\omega\varepsilon'}A_n\label{eqn:EnormTM}\\
H_n&=0\label{eqn:HnormTM}\\
\vec{E}_\bot&=\frac{1}{j\omega\varepsilon'}\vec{\nabla}_\bot\frac{\partial A_n}{\partial n}\label{eqn:EperpTM}\\
\vec{H}_\bot&=\vec{\nabla}_\bot{A_n}\times\hat{n}\label{eqn:HperpTM}
\end{align}

\subsection{$TE_n$ Solutions $(E_n=0)$}
It is apparent by examining (\ref{eqn:Enorm0}) and (\ref{eqn:Hnorm0}) that in order to obtain $TE_n$ solutions then the following conditions must be met.
\begin{align}
	\vec{A} = 0\\
	\vec{F} = F_n\hat{n}
\end{align}
and then the fields are found form (\ref{eqn:Enorm1})--(\ref{eqn:Hperp1}) to be,
\begin{align}
E_n&=0\label{eqn:EnormTE}\\
H_n&=\frac{-(\gamma_\bot^2=-k_\bot^2)}{j\omega\mu'}F_n\label{eqn:HnormTE}\\
\vec{E}_\bot&=-\vec{\nabla}_\bot{F_n}\times\hat{n}\label{eqn:EperpTE}\\
\vec{H}_\bot&=\frac{1}{j\omega\mu'}\vec{\nabla}_\bot\frac{\partial F_n}{\partial n}\label{eqn:HperpTE}
\end{align}